In 1869 Mendeleev introduced his 'periodic table of the elements'. With this table, Mendeleev organised all the different chemical properties of the atom species. Gaps in the table indicated atoms that still had to be discovered. More than 100 years after Mendeleev focused on atoms, new periodic tables have been constructed for other types of 'objects'. At present the list of periodic tables includes e.g. a table for fundamental particles (the standard model), for hadrons (the eightfold way), for atom nuclei (the isotope chart) and for organisms (the tree of life). The operator hierarchy now suggests a connection between these tables, because each of the existing tables organises its proper type of 'elements' (atoms, molecules, bacteria, etc.), and every existing periodic table condenses to a single 'box' in the operator hierarchy. The result is more than a 'periodic table' for periodic tables, because the steps are linked by means of topological mechanisms. To show how this works, periodic tables are introduced below in order of increasing complexity of the elements. At the end, an overview is provided which allows the indication of 'missing periodic tables'.

For more references and information about this topic:General laws and centripetal science.
Jagers op Akkerhuis G.A.J.M. (2014). European Review 22: 113-144

A periodic table of the fundamental particles: The standard model

Probably the most fundamental periodic table is the ‘standard model’ used in particle physics. It categorises the major classes of fundamental particles as either force carrying particles (bosons) or matter particles (fermions). The fermions are subsequently divided into leptons or quarks, both of which are partitioned over three groups of increasing mass.

File:Standard Model of Elementary Particles.svg

A periodic table of the hadrons.

Another fundamental periodic table used in particle physics is the ‘eightfold way’. This table is used to organise the many ways by which quarks can combine into hadrons. Hadrons consisting of two quarks are called mesons while those made up of three quarks are called baryons ('baryos' means 'heavy'), and a separate table exists for each of these types. The eightfold way was developed by Gell-Mann and Nishijima and received important contributions from Ne'eman and Zweig (Gell-Man and Neeman 1964).

The periodic table of the elements (Mendeleev 1869)

The most well-known periodic table is the periodic table of the elements. Mendeleev introduced this tabular display of the chemical elements in 1869. It organised the reactivity of atoms and indicated a number of missing elements. Mendeleev’s discovery was so important that his table is still used as a basic tool in chemistry. Furthermore, two tables can be considered the fundaments of Mendeleev’s periodic table: the ‘nucleotide chart’ and the charts showing which sets of electron shells are to be expected in relation to a given number of protons.

A periodic table for the descendence of all organsm types: The tree of life

Finally, and even though it may be a bit unusual to regard this arrangement as a periodic table, there are also good grounds to include the ‘‘tree of life” in this overview of tabular presentations. The only difference with the other tables is that the tree of life includes descent, a property that has no meaning in the other periodic tables discussed so far. In all other aspects, the tree of life has similar properties of creating a unique and meaningful overview of all basal types of operators, which enter the scheme as species.

A periodic table of periodic tables

Every single periodic table discussed above is central to its proper field of science. But the tables are not connected. The If one uses the operator hierarchy, it becomes possible to connect the separate tables by focusing on how the elements in a previous table give rise to the elements in the next table. Because of this mechanistic connection, the operator hierarchy can be used as a ‘periodic table for periodic tables’ that organises the elements of all existing periodic tables. In fact, the operator hierarchy indicates that while there exists a nucleotide chart, there is no chart at one higher level for the autocatalytic sets. Also other periodic tables seem to be lacking, e.g. for quark-gluon hypercycles, for membrane types or for types of sensors.

A periodic table for periodic tables that is based on the operator hierarchy. Grey rows represent operators.

The operator hierarchy. Grey columns represent dimensions that are not operators.

Discussion

The above inventory of periodic tables demonstrated the existence of a periodic table for almost every complexity level in the operator hierarchy. The inventory furthermore suggested gaps in the operator scheme, for which no periodic tables were found: the quark-gluon hypercycles, the quark confinement, the molecules, the autocatalytic sets, the cellular membranes, the cyclic CALM networks and the sensory interfaces. With the exception of the molecules, which may not have a periodic table because of the almost unlimited number of combinations that can be made from the various atom species, all the missing tables involve hypercyclic sets and interfaces. One may now suggest that it is generally impossible to create periodic tables for hypercyclic sets or for interfaces, but this assumption is at least partially contradicted by the nucleotide chart and the classification of electron shells. A reason for the absence of tables for hypercyclic sets may be that the number of possible configurations is so large that it is impossible to classify them, in the same way that it is hard to classify molecular configurations. Such ideas, however, need to be worked out in more detail.